Solve For B B B : − 18 + B ≤ − 5 B + 4 ( − 2 B + 20 -18 + B \leq -5b + 4(-2b + 20 − 18 + B ≤ − 5 B + 4 ( − 2 B + 20 ]Write Your Answer With B B B First, Followed By An Inequality Symbol.

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear inequality, which is given as: $-18 + b \leq -5b + 4(-2b + 20)$. Our goal is to isolate the variable $b$ and express the solution in the form of an inequality.

Understanding the Inequality

Before we dive into solving the inequality, let's first understand what it represents. The given inequality is a linear inequality, which means it is an inequality that can be written in the form of $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants. In this case, the inequality is $-18 + b \leq -5b + 4(-2b + 20)$.

Step 1: Simplify the Right-Hand Side

To solve the inequality, we need to simplify the right-hand side by evaluating the expression inside the parentheses. We have $4(-2b + 20)$, which can be simplified as follows:

$$4(-2b + 20) = -8b + 80$$

Now, the inequality becomes:

$$-18 + b \leq -5b - 8b + 80$$

Step 2: Combine Like Terms

Next, we need to combine like terms on the right-hand side. We have $-5b$ and $-8b$, which can be combined as follows:

$$-5b - 8b = -13b$$

Now, the inequality becomes:

$$-18 + b \leq -13b + 80$$

Step 3: Add $13b$ to Both Sides

To isolate the variable $b$, we need to add $13b$ to both sides of the inequality. This will give us:

$$-18 + 13b + b \leq -13b + 13b + 80$$

Simplifying the left-hand side, we get:

$$-18 + 14b \leq 80$$

Step 4: Subtract 80 from Both Sides

Next, we need to subtract 80 from both sides of the inequality. This will give us:

$$-18 + 14b - 80 \leq 80 - 80$$

Simplifying the left-hand side, we get:

$$-98 + 14b \leq 0$$

Step 5: Add 98 to Both Sides

To isolate the variable $b$, we need to add 98 to both sides of the inequality. This will give us:

$$-98 + 98 + 14b \leq 0 + 98$$

Simplifying the left-hand side, we get:

$$14b \leq 98$$

Step 6: Divide Both Sides by 14

Finally, we need to divide both sides of the inequality by 14. This will give us:

$$\frac{14b}{14} \leq \frac{98}{14}$$

Simplifying the left-hand side, we get:

$$b \leq \frac{98}{14}$$

Simplifying the Right-Hand Side

To simplify the right-hand side, we can divide 98 by 14, which gives us:

$$\frac{98}{14} = 7$$

Therefore, the final solution to the inequality is:

$b \leq 7$

Conclusion

In this article, we have solved a linear inequality of the form $-18 + b \leq -5b + 4(-2b + 20)$. We have followed a step-by-step approach to isolate the variable $b$ and express the solution in the form of an inequality. The final solution is $b \leq 7$, which means that the value of $b$ must be less than or equal to 7.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations (PEMDAS) and to simplify the right-hand side by evaluating expressions inside parentheses. Additionally, be sure to combine like terms and add or subtract the same value to both sides of the inequality.

Common Mistakes to Avoid

When solving linear inequalities, some common mistakes to avoid include:

• Not following the order of operations (PEMDAS)
• Not simplifying the right-hand side by evaluating expressions inside parentheses
• Not combining like terms
• Not adding or subtracting the same value to both sides of the inequality

Introduction

In our previous article, we solved a linear inequality of the form $-18 + b \leq -5b + 4(-2b + 20)$. We followed a step-by-step approach to isolate the variable $b$ and express the solution in the form of an inequality. In this article, we will answer some common questions related to solving linear inequalities.

Q: What is a linear inequality?

A linear inequality is an inequality that can be written in the form of $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I solve a linear inequality?

To solve a linear inequality, you need to follow these steps:

1. Simplify the right-hand side by evaluating expressions inside parentheses.
2. Combine like terms on the right-hand side.
3. Add or subtract the same value to both sides of the inequality.
4. Divide both sides of the inequality by a non-zero constant.

Q: What is the difference between a linear inequality and a linear equation?

A linear equation is an equation that can be written in the form of $ax + b = cx + d$, where $a$, $b$, $c$, and $d$ are constants. A linear inequality, on the other hand, is an inequality that can be written in the form of $ax + b \leq cx + d$ or $ax + b \geq cx + d$.

Q: How do I know which direction to add or subtract when solving a linear inequality?

When solving a linear inequality, you need to add or subtract the same value to both sides of the inequality. If the inequality is of the form $ax + b \leq cx + d$, you need to add the same value to both sides. If the inequality is of the form $ax + b \geq cx + d$, you need to subtract the same value from both sides.

Q: What is the difference between a strict inequality and a non-strict inequality?

A strict inequality is an inequality that is written with a strict inequality symbol, such as $ax + b < cx + d$ or $ax + b > cx + d$. A non-strict inequality, on the other hand, is an inequality that is written with a non-strict inequality symbol, such as $ax + b \leq cx + d$ or $ax + b \geq cx + d$.

Q: How do I graph a linear inequality on a number line?

To graph a linear inequality on a number line, you need to follow these steps:

1. Draw a number line and mark the point that corresponds to the value of $x$ that makes the inequality true.
2. Draw an arrow on the number line to indicate the direction of the inequality.
3. Shade the region on the number line that corresponds to the solution set of the inequality.

Q: What is the solution set of a linear inequality?

The solution set of a linear inequality is the set of all values of $x$ that make the inequality true.

Q: How do I find the solution set of a linear inequality?

To find the solution set of a linear inequality, you need to solve the inequality and express the solution in the form of an inequality.

Conclusion

In this article, we have answered some common questions related to solving linear inequalities. We have discussed the difference between a linear inequality and a linear equation, the difference between a strict inequality and a non-strict inequality, and how to graph a linear inequality on a number line. We have also discussed how to find the solution set of a linear inequality. By following these tips and techniques, you can become proficient in solving linear inequalities and apply this skill to a wide range of mathematical problems.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations (PEMDAS) and to simplify the right-hand side by evaluating expressions inside parentheses. Additionally, be sure to combine like terms and add or subtract the same value to both sides of the inequality.

Common Mistakes to Avoid

When solving linear inequalities, some common mistakes to avoid include:

• Not following the order of operations (PEMDAS)
• Not simplifying the right-hand side by evaluating expressions inside parentheses
• Not combining like terms
• Not adding or subtracting the same value to both sides of the inequality

By following these tips and avoiding common mistakes, you can become proficient in solving linear inequalities and apply this skill to a wide range of mathematical problems.